The following article has been excerpted from Math Education for Gifted Students, one of six exciting books in the Gifted Child Today Reader Series. This series brings together the best articles published in Gifted Child Today, the nation's most popular gifted education journal. Each book in the series is filled with exciting and practical classroom ideas, useful summaries of research findings, and discussions of identification and classroom management, and informed opionion about educating gifted children.

Chapter 6

What does effective differentiation look like in a math classroom, and in particular, what does it look like in a mixed-ability math classroom? Those essential questions must be confronted by teachers and program directors who work with gifted and talented students in the field of mathematics. Once a commitment is made, it is not acceptable for students with high abilities in math to traverse lethargically the terrain of the mathematics curriculum. Educators of the gifted and talented must confront the best practices and ask, “How can we apply effective differentiation practices to meet the needs of our students?”

The following is a brief summary of what current research reveals about mathematics instruction with gifted learners. We then recommend a model that teachers can use in mixed-ability classrooms to challenge students effectively and raise their achievement in mathematics.

Particularly in mathematics, research supports the academic effects of accelerated or advanced curricula for high-ability learners (VanTassel-Baska & Brown, 2005). Additionally, Kulik and Kulik’s (1992) meta-analysis supported the notion that acceleration, when used in tandem with ability grouping, has stronger effects on student learning than enrichment. Similarly, Walberg (1991), in comparing acceleration to other strategies (e.g., independent study, various modes of grouping, and problem solving) found that acceleration showed powerful treatment effects and differentiated educational gains.

Research on the acceleration of mathematical strategies supports the use of increased pace and less repetition for advanced learners. In an investigation of a group of students in grades 7–12 identified as having high abilities in mathematics, Kolitch and Brody (1992) found that these students were successful when they took mathematics courses several years earlier than their same-age peers. In this study, students took calculus an average of 2.5 years before the pace suggested by traditional high school programs. In following these students, the researchers found that achievement in the accelerated calculus course and subsequent postcalculus courses was high. Students reported no difficulty or minimal difficulty in this type of acceleration, and most completed college math courses while still in high school.

Mills, Ablard, and Lynch (1992) investigated the effectiveness of shortened, accelerated mathematics courses with students identified as mathematically talented. Their study documented the use of individually paced summer courses by examining the students’ success in their successive courses. The summer courses provided students the opportunity to complete entire courses in algebra, precalculus, and even calculus. After being accelerated during the summer program, students reported feeling well prepared for the courses following the summer course, and, in the majority of cases, the students maintained A averages.

Acceleration benefits are not specific to mathematical content. Lynch (1992) documented the effectiveness of “fast-paced” high school science courses for gifted students. The Center for Gifted Education at the College of William and Mary has conducted research that follows up on the effectiveness of accelerated pacing and advanced content for all subject areas (VanTassel-Baska, Avery, Little, & Hughes, 2000; VanTassel-Baska, Bass, Ries, Poland, & Avery, 1998; VanTassel-Baska, Johnson, Hughes, & Boyce, 1996). This research helps to provide support for content acceleration and pacing of academic content.

Gifted learners have cognitive differences that necessitate a differentiated approach to curricular and instructional practices within mathematics classrooms. One primary difference is gifted students’ ability to acquire new and complex information more rapidly than their average-ability peers (Feldhusen, 1994; Sternberg, 1988).

Rogers (2000) revealed three vital insights into the cognitive difference of gifted learners in mathematics classrooms. First, the rate at which children with intelligence scores above 130 learn and process information is approximately eight times faster than children with intelligence scores below 70 (i.e., students identified as having serious mental deficiencies). Second, students with high abilities in mathematics are more likely to retain science and mathematics content with almost complete accuracy when taught two or three times faster than the average class pace. Third, students with high abilities in math are more likely to forget or mislearn science and mathematics content when they review it more than two or three times. Taking these points into consideration, students with high abilities in mathematics will learn mathematical concepts with fewer repetitions than their average-ability peers. This difference presents a distinct challenge to the math teacher. More specifically, how can the needs of all students be met in one classroom?

One answer to this challenge can be found in a mixed-ability classroom, where questions arise such as “How does one manage an instructional program where a small group of students needs one or two repetitions of a new concept while the majority of the class needs four or five repetitions?” In reality, it is difficult to accelerate a cluster of students while maintaining a typical pace that uses multiple repetitions for the rest of the class. There are several approaches to accelerating and differentiating instructional and curricular strategies for advanced students, including telescoping the curriculum and tiered objectives.

Telescoping is curriculum compacting. It may involve compacting up to 3 years of the curriculum into 2 academic years or 2 years of the curriculum into 1 academic year. Telescoping is a logical approach to mathematics differentiation because of the linear nature of the skills in the curriculum, and telescoping has produced documented achievement gains for students at the secondary level (Lynch, 1992; Mills et al., 1992).

As promising as the benefits of telescoping appear, it still presents instructional difficulty in terms of managing a flexible pace in a mixed-ability classroom. However, a tiered-objectives model can provide a manageable and effective method for telescoping the mathematics curriculum in a mixed-ability setting (Tomlinson, 1999).

Tiered objectives and tiered activities are ways for teachers to ensure that students work at appropriate levels of challenge while studying the same essential skills and concepts. The guidelines for developing tiered activities include four different parts (Tomlinson, 1999):

1. Identify the objectives, concepts, and skills to be taught.
2. Create a set of activities to teaching that objective for students working on grade level.
3. Identify the next level of increasing complexity. Then, develop a set of activities to teach the same concept at an increased level of complexity. This same step can be replicated for more than one tier if desired.
4. Group students according to assessed levels of readiness and assign them to the set of activities for the different tiers.

The practice of tiered mathematics objectives has two underlying assumptions. First, students with high abilities in mathematics will learn new concepts with fewer repetitions than their peers. Second, mathematics objectives are logically sequenced by grade level and concepts are replicated in successive years with increasing difficulty. Teachers sometimes think of telescoping as a linear process, rather than a layered process.

When envisioned as a linear process, teachers and program directors understand acceleration as occurring in a specific, preestablished progression. In other words, if asked to complete a two-in-one telescope of the curriculum, a sixth-grade teacher may imagine teaching all the sixth-grade objectives before Christmas and all the seventh-grade objectives between January and Memorial Day. If asked to complete a three-in-two telescope, she may imagine completing the sixth-grade curriculum and then completing the first half of the seventh-grade curriculum in the last 2 or 3 months of the academic year. However, this view is not congruent with what works best for teachers or students.

When the practice of tiered objectives is viewed as a layered telescoping model, rather than a linear telescoping model, the general curriculum map of the academic year is based upon completing one grade-level set of learner objectives (e.g., all sixth-grade objectives). The telescoping occurs during the repetitions of these objectives. Advanced students learn each concept or skill at 2 years’ worth of complexity in the same number or repetitions a typical learner acquires the grade-level knowledge. As the typical learner will need four or five repetitions to master a concept, the gifted learner masters the grade-level objective in one or two. While the majority of the class is still working on the grade-level expectation of a given concept, the gifted learner is solving problems based on the increased complexity of a tiered objective of the concept (e.g., seventh-grade objective, rather than sixth). When this model is applied to all objectives in a content area, it is possible for a student to master the curriculum expectations of two grade levels in one year, even given the management complexities of a mixed-ability classroom.

The following example uses sixth-grade mathematics as a starting point. Table 6.1 presents the parallel sequence (based on the Texas Essential Knowledge and Skills) of how the learner objectives increase in complexity from grades 6–8.

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After introducing the concept of angle measurement and modeling how to measure angles, the teacher could have students work in small groups to measure angles related to objects around the room and label the angles with the appropriate classification according to their angle measures. After the small-group practice, individual students could be given a set of different triangles or quadrilaterals. As the rest of the class measures the angles and predicts a relationship, students with high abilities in mathematics may work independently to master the necessary concepts and terms, while other students benefit from repetitions with the original content. Students with high abilities in mathematics can be challenged and accelerated by applying angle measurement to concepts that would normally be taught in the seventh-grade curriculum, such as complementary and supplementary angles, as well as similar figures.

In gaining mastery of these higher grade-level objectives, students could use geometry exploration software or pencils and paper to set up a diagram of two adjacent angles that form a complementary (right) or a supplementary (straight) angle. Students may also manipulate the common ray in the interior of the right angle and straight angle to record the resulting measurement of each independent angle (see Figure 6.1). Equations of these measurement pairs could be displayed in a table. If students master this concept quickly, then the teacher can continue to accelerate them with eighth-grade concepts, such as exploring the relationship between corresponding angles in a variety of geometric figures.

Educators should provide all learners with opportunities to obtain optimal levels of learning. Many, if not most, classrooms include learners with mixed abilities. Particularly in mathematics classes, these learner differences may be significant. In order to attain optimal levels of learning for all students, instructional leaders must move beyond the one-size-fits-all conception of curricular and instructional practices. Rather, the curriculum should include a sequence of learning activities that is constantly being developed in response to learner readiness, which includes the point at which a student enters a particular study and the pace at which he or she acquires new knowledge and skills.

Differentiating curricular and instructional practices requires that teachers modify the depth and complexity of concepts, as well as the pace of instruction. Current best practice research indicates that acceleration strategies, such as working with above-grade-level objectives and completing multiple levels of curricula in a year, provides significant academic benefits. Applying tiered objectives accomplishes the principles and benefits of acceleration while maintaining a manageable framework for teachers.

Teachers or program coordinators interested in implementing tiered objectives as an accelerated program option need opportunities to work with content-based vertical teams to align curricular objectives. This can be accomplished by having all teachers in a subject or content area, regardless of grade level, identify where the objectives meet and overlap. Tiering objectives occurs when teachers can easily access and identify the objectives across grade levels that are sequenced with those they personally teach. It is recommended that teachers in vertical teams develop objective charts similar to the example in Table 6.1 so that objectives can be conceptualized as being fluid. Doing so will allow teachers to think about teaching concepts in a field of study, rather than teaching Chapter 7 in the textbook. If teachers are unable to explore objectives as interrelated, they may not quickly see how the concepts in Chapter 7 in the sixth-grade textbook correspond to the same concept taught at a higher level of complexity in Chapter 5 of the seventh-grade textbook. Allowing teachers to work in vertical teams encourages them to teach concepts, rather than chunks of knowledge and skills.

School curricular documents could include not only an aligned sequence of concepts and objectives, but also information about grade-level resources such as textbooks that are available to teach specific objectives. Creating a reference system in this manner provides teachers with resources (such as texts and materials) at a variety of grade levels. Having such resources available increases teachers’ abilities to meet the needs of all students. Unfortunately, teachers often report that their campus administrators or department chairs will not allow them to use textbooks from other grade levels. Access to a variety of resources allows every student to have an opportunity to achieve his or her optimal level of learning. Thus, teachers must have access to the resources that will assist student learning.

The tiered objectives model is guided by the principle that teachers can teach one concept to the whole class, while students develop knowledge and skills related to that concept at different levels of complexity. The essence of this strategy is undermined if teachers are not equipped with proper resources to teach concepts at each student’s point of entry.

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Kolitch, E. R., & Brody, L. E. (1992). Mathematics acceleration of highly talented students: An evaluation. Gifted Child Quarterly, 36, 78–86.

Kulik, J. A., & Kulik, C. C. (1992). Meta-analytic findings on grouping programs. Gifted Child Quarterly, 36, 73–77.

Lynch, S. J. (1992). Fast-paced high school science for academically talented: A 6-year perspective. Gifted Child Quarterly, 36, 147–154.

Mills, C. J., Ablard, K. E., & Lynch, S. J. (1992). Academically talented students’ preparation for advanced-level coursework after individually-paced precalculus class. Journal for the Education of the Gifted, 16, 2–17.

Rogers, K. B. (2000, December). Lessons learned in the 20th century to help us in the 21st. Paper presented at the annual meeting of the Texas Association for the Gifted and Talented, Austin.

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VanTassel-Baska, J., & Brown, E. F. (2005). An analysis of gifted education curriculum models. In F. A. Karnes & S. M. Bean (Eds.), Methods and materials for teaching the gifted (2nd ed., pp. 75–106). Waco, TX: Prufrock Press.

VanTassel-Baska, J., Johnson, D. T., Hughes, C. E., & Boyce, L. N. (1996). A study of language arts curriculum effectiveness with gifted learners. Journal for the Education of the Gifted, 19, 461–480.

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